Fourier Transform in L

ISSN 2219-7184; Copyright ICSRS Publication, 2010c www.i-csrs.org

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Fourier Transform in L

(R) Spaces, p ≥ 1

Devendra Kumar and Dimple Singh Department of Mathematics

Research and Post Graduate Studies

M.M.H.College, Model Town, Ghaziabad-201001, U.P.India E-mail:d [email protected]

(Received: 17-11-10/ Accepted: 31-12-10) Abstract

A method for restricting the Fourier transform of f ∈L^p(R),1≤p≤ ∞, spaces have been discussed by using the approximate identities.

Keywords: Approximate identities, convolution operator, Schwartz space and atomic measure.

1 Introduction

Letf ∈ L¹(R).The Fourier transform of f(x) is denoted by ˆf(ξ) and defined by

f(ξ) =ˆ 1

√2π Z

R

f(x)e^−iξxdx, ξ∈R. (1) Iff ∈L¹(R) and ˆf ∈L¹(R), then the inverse Fourier transform of ˆf is defined by

f(x) = 1

√2π Z

R

f(ξ)eˆ ^iξxdξ (2)

for a.e. x∈R. If f is continuous, then(1.2) holds for everyx.

It is known that several elementary functions, such as constant function, sinwt, coswt, do not belongs to L¹(R) and hence they do not have Fourier transforms. But when these functions are multiplied by characteristic function, the resulting functions belongs to L¹(R) and have Fourier transforms. Many

applications, including the analysis of stationary signals and real time signal processing, make an effective use of Fourier transform in time and frequency domains.

The remarkable success of the Fourier transform analysis is due to the fact that, under certain conditions, the signal can be reconstructed by the Fourier inversion formula. Thus the Fourier transform theory has been very useful for analyzing harmonic signals or signals for which there is no need for local information. On the other hand, Fourier transform analysis has also been very useful in many other areas, including quantum mechanics, wave motion and turbulence.

By Lebesgue lemma we have if f ∈ L¹(R) then lim|ξ|→∞|fˆ(ξ)| = 0, it fol- lows that Fourier transform is a continuous linear operator from L¹(R) into Co(R), the space of all continuous functions on R which decay at infinity, that is, f(x) → 0 as |x| → ∞. Roughly we say that if f ∈ L¹(R), it does not necessarily imply that ˆf also belongs toL¹(R).

Bellow [1] and Reinhold - Larsson [2] constructed examples of sequence of natural numbers along which the individual ergodic theorem holds in someL^p spaces (good behavior) and not in others (bad behaviour). In particular, well behaved sequences were perturbed in such a way that good behavior persists only in certain spaces.

In the present work we provide a method for restricting the Fourier trans- form of f ∈ L^p(R) spaces using the pointwise convergence of convolution operators for approximate identities.

Definition 1.1. Let ϕ ∈ L¹(R) such that ˆϕ(0) = 1. Then ϕ_ε(x) = ε⁻¹ϕ(x/ε) is called an approximate identity if

(i)R

Rϕ_ε(x)dx= 1 (ii) sup_ε>0R

R|ϕ_ε(x)|dx <+∞, (iii) limε→0

R

|x|>δ|ϕε(x)|dx= 0,for all δ >0.

Proof. Properties (i) and (ii) can be proved by observing Z

R

ϕ_ε(x)dx = Z

R

ε⁻¹ϕ(x/ε)dx= Z

R

ϕ(x/ε)d(x/ε) = 1.

For (iii), we have Z

|x|>δ

ϕ_ε(x)dx= Z

|x|>δ

1

εϕ(x/ε)dx= Z ∞

δ

1

εϕ(x/ε)dx+ Z −δ

−∞

1

εϕ(x/ε)dx.

Substitutingy=x/ε, we get

limε→0

Z ∞

δ/ε

ϕ(y)dy+ Z −δ/ε

−∞

ϕ(y)dy= 0.

Definition 1.2. A sequence of functions {φn}n∈N such that φn(x) = nφ(nx) where n = ¹_ε, n → ∞, ε −→ 0 is called an approximate identity if

(i)R

Rφn(x)dx= 1 for all n, (ii)sup_nR

R|φ_n(x)|dx <+∞, (iii)limn→∞

R

|x|>δ|φ_n(x)|dx= 0 for everyδ >0.

In the consequence of above Definition 1.2, we can easily prove the follow- ing proposition.

Proposition 1.1. A sequence of functions{φ_n}n∈N with φ_n ≥0,φˆ_n(0) = 1 is an approximate identity if for every ε > 0 there exists n_o ∈ N so that for alln≥no we have Rε

−εφn >1−ε.

Let us consider the class S(R) of rapidly decreasing C^∞−functions on R i.e., Schwartz class such that

S(R) = {f :R→R,sup

x∈R

(xⁿ d^m

dx^mf)(x)<∞}n, m∈N ∪(0).

It is well known that iff ∈S(R) then ˆf ∈ S(R) and S(R)⊂L^p(R).To prove the denseness ofS(R)∈L^p(R), we have

ρ∈S(R)⇒ |ρ(x)| ≤ c 1 +|x|ⁿ. For 1≤p <∞,

Z

R

|ρ(x)|^pdx≤ Z

R

c^p

(1 +|x|ⁿ)^p <∞

which gives ρ∈L^p(R). Define a sequence {ρ_N} such that

ρ_N(x) =

f(x), if −N ≤x≤N; 0, otherwise ;

⇒ ∃ρ_N ∈S(R), f ∈L^p(R) such that Z

R

|ρ_N −f|^pdx→0 asN → ∞. Hence S(R) is dense in L^p(R).

Remark 1.1. If 0≤φ(x)∈S(R) and ˆφ(0) = 1. Then φ_n(x) = nφ(nx) is an approximate identity.

Proposition 1.2. If f ∈L¹(R) and φ ∈S(R) thenφ∗f ∈S(R).

Proof. We have

φ∗f = Z

R

φ(y)f(x−y)dy

dⁿ

dxⁿ(φ∗f) = Z

R

φ(y) dⁿ

dxⁿf(x−y)dy or

|x|ⁿ dⁿ

dxⁿ(φ∗f) =|x|ⁿ Z

R

f(x−y) dⁿ

dyⁿφ(y)dy, substitutingx−y=z, we obtain,

= Z

R

f(y)|x|ⁿ dⁿ

dxⁿφ(x−y)dy using|x−y| ≤ |x|+|y| ≤ ^3|x|₂ , we get

= Z

|y|>^|x|₂

f(y)|x|ⁿ dⁿ

dxⁿφ(x−y)dy+ Z

|y|≤^|x|₂

f(y)|x|ⁿ dⁿ

dxⁿφ(x−y)dy→0.

Proposition 1.3. Ifφn(x) is an approximate identity andf ∈L^p(R) then φ_n∗f →f ∈L^p(R).

Proof. Consider [

Z

R

|(φ_n∗f)(x)−f(x)|^pdx]^1/p = [ Z

R

dx|

Z

R

φ_n(x−y)f(y)dy−f(x)|^p]^1/p

= [ Z

R

dx|

Z

R

φ_n(y)f(x−y)dy−f(x)|^p]^1/p

usingf(x) =R

Rf(x)φ_n(y)dy in above we obtain [R

Rdx|R

Rφn(y)(f(x−y)−f(x))dy|^p]^1/p

≤ [ Z

R

dx Z

|y|>δ

|φ_n(y)|^p|f(x−y)−f(x)|^pdy]^1/p + [

Z

R

dx Z

|y|≤δ

|φn(y)|^p.|f(x−y)−f(x)|^p]^1/p (3)

≤ Z

|y|>δ

dy|φ_n(y)|[

Z

R

dx|f(x−y)−f(x)|^p]^1/p +

Z

|y|≤δ

dy|φ_n(y)|[

Z

R

|f(x−y)−f(x)|^pdx]^1/p (4)

≤ Z

|y|>δ

dy|φ_n(y)|(2kf k_p) + Z

|y|≤δ

dy|φ_n(y)|sup

|y|≤δ

[ Z

R

|f(x−y)−f(x)|^pdx]^1/p.

Proceeding limits asn → ∞, the right hand side tends to zero since sup

|y|<δ

[ Z

R

|f(x−y)−f(x)|^pdx]^1/p →0.

Hence the proof is completed.

Proposition 1.4. Let φ_n=α_nϕ_n+ (1−α_n)σ_n,where {ϕ_n}n∈N, {σ_n}n∈N

are approximate identities and 0≤α_n≤1.

(a) For 1 ≤ p < +∞ and every f ∈ L^p(R), limn→∞(φ_n−ϕ_n)∗f → 0 and limn→∞(φ_n−σ_n)∗f →0.

(b)For every f ∈L^∞(R), limn→∞(φ_n−ϕ_n)∗f →0 a.e. . (c) For 1 ≤ p < ∞, if P

n(1 − α_n)^p < +∞, then for every f ∈ L^p(R), limn→∞(φ_n−ϕ_n)∗f →0 a.e. .

Proof.(a) Set 1 ≤p≤ ∞, andf ∈L^p(R).In view of Minkowski’s inequal- ity

k(φn−ϕn)∗f kp≤(1−αn)(kσn∗f −f kp +kϕn∗f −f kp) and using Proposition 1.3 we obtain k(φ_n−σ_n)∗f k_p→0.

(b)Forf ∈L^∞(R), |(φ_n−ϕ_n)∗f| ≤k(φ_n−ϕ_n)∗f k→0 by part(a).

(c)Forf ∈L^p(R) Z

R

X

n

(1−α_n)^p|σ_n∗f(x)|^pdx = X

n

k(1−α_n)σ_n∗f k^p_p

≤ X

n

(1−σn)^p kf k^p_p<+∞.

Then (1−α_n)σ_n∗f →0 a.e. . Similarly (α_n−1)ϕ_n∗f →0 a.e. .

Definition 1.3. An approximate identity {φ_n} is called L^p− good if φ_n ∗f → f a.e. for all f ∈ L^p(R), and it is called good if it is L^p−good for every 1 ≤ p ≤ +∞. An approximate identity {φ_n} is called L^p−bad if there exists f ∈L^p(R) such that φ_n∗f 9f on a set of positive measure.

Definition 1.4. Let {ϕ_n}_n∈N and {σ_n}_n∈N be approximate identities, α_n be a sequence of real numbers with 0 ≤ α_n ≤ 1 and α_n → 1. We call per- turbed approximate identities any approximate identity{φ_n}n∈N of the form φ_nϕ_n+ (1−α_n)σ_n.

2 Main Results

Theorem 2.1.

(i)Given any good approximate identity {ϕ_n}n∈N there exists a perturbed approximate identity {φn}n∈N such thatf ∈L^q(R)

(φn∗ˆf)(ξ) = ˆφn(ξ) ˆf(ξ) ˆ

φˆ_n(ξ) ˆf(ξ)

→f(x) forq ≥p, p∈[1,∞) and

( ˆφ_n(ξ) ˆˆf(ξ))9f(x) for 1≤q < p.

(ii) ˆ

( ˆφ_n(ξ) ˆf(ξ))→f(x) forq > p and ( ˆφ_n(ξ) ˆˆf(ξ))9f(x) for 1 ≤q ≤p.

(iii) ˆ

( ˆφ_n(ξ) ˆf(ξ))→f(x) forq=∞

( ˆφ_n(ξ) ˆˆf(ξ))9f(x) for 1 ≤q <∞.

Proof. (i) Let

g_n(x) = 1

√2π Z

R

e^ixξφˆ_n(ξ) ˆf(ξ)dξ

= 1

√2π Z

R

e^ixξ φˆ_n(ξ)

√2π Z

R

e^−iξyf(y)dydξ

= 1

2π Z

R

e^i(x−y)ξφˆ_n(ξ) Z

R

f(y)dy

= 1

√2π Z

R

φ_n(x−y)f(y)dy or = (φ_n∗f)(x)

( ˆφ_n(ξ) ˆˆf(ξ)) = 1

√2π Z

R

e^ixξφˆ_n(ξ) ˆf(ξ)dξ = (φ_n∗f)(x).

Fixq ≥p and taking 1−α_n = _(nlog¹2n)^1/p Since Σ_n(1−α_n)^q <+∞ and ϕ_n is anL^q−good approximate identity, using Proposition 1.4 we obtain that {φ_n} is also an L^q−good approximate identity.

Hence for q≥p,(φ_n∗f)(x)→f(x).

Now we have to prove that for each 1 ≤ q < p. there exists f_q ∈ L^q(R) so that lim sup_κ|x|^{κ d}_dx^κκ(φ_κ∗f_q → ∞) on a set of positive measure.

Set

f_q(x) = 1

(xlog²(x/2))^1/qχ_[0,1](x)∈L_q(R).

Choose

rn = 1

n^1+1/p(logn)^2/p, an =r

1 p+1

n = 1

n^1/p(logn)^p(p+1)2 ,

J_n = [a_n−r_n, a_n+r_n] and

U_n= [−a_n+r_n,−a_n+1+r_n+1], for sufficiently large n and for allκ≥n,x∈U_κ,

φ_κ∗f_q(x) ≥ (1−α_κ)σ_κ∗f_q(x)

≥ 1

(κlog²κ)^1/p Z

−J_κ

σ_κ(y)f_q(x−y)dy.

Now, we get

φ_κ∗f_q(x)≥ f_q(C_rκ(logκ)^2/p+1) (κlog²κ)^1/p

Z

−Jκ

σ_κ(y)dy or

f_q(C_rκ(logκ)^2/p+1) = κ^1/q+1/pq(logκ)^pq(p+1)2

C^1/q(log(C/2κ^(p+1)/p(logκ)^2/p(p+1)))^2/q. Then

φ_κ∗f_q(x)≥Cκ^1q−p1+pq1 H_q(κ)> κ^δ ≥n^δ, where

H_q(κ) = (logκ)^{pq(p+1)2 −2p}

C^1/q(logC/2κ^(p+1)/p(logκ)^2/p(p+1))^2/q and

0< δ <1/q−1/p+ 1/pq.

So

d^κ

dx^κ(φ_κ∗f_q(x))≥C d^κ

dx^κ(κ1/q−1/p+1/pq

H_q(κ)) or

|x|ⁿ dⁿ

dxⁿ(φ_n∗f_q(x))≥ |x|ⁿ Z

−Jκ

f_q(x−y) dⁿ

dyⁿσ_κ(y)dy forκ≥n

|x|^κ d^κ

dx^κ(φκ ∗fq(x))≥ |x|ⁿ dⁿ

dxⁿn^δ ≥ |x|ⁿ dⁿ

dxⁿ( 1 (x−y)^pδ)

= |x|ⁿ(−1)ⁿ(pδ+n−1)!

(pδ)!(x−y)^pδ+n

≥ |x|ⁿ (−1)ⁿ(pδ+n−1)!

(pδ)!Cr_n(logn)^2/p+1(logn)^2δ/p+1

→ ∞ asn → ∞.

In view of Sawyer’s Principle [3] there exists a functionsf ∈L^q([0,1)) ⊆L^q(R) such that lim sup_n|x|^{n d}_dxⁿn(φ_n∗f)→ ∞a.e. on a set of positive measure in R, It follows that φ_n∗f not belongs to S(R) or φ_n∗f 9f orφˆ_n(ξ) ˆˆf(ξ)9 f(x)

for 1≤q < p.

(ii) Let pn be a decreasing sequence of real numbers such that p1 > p2 >

....p_n > ... & p. for each p_i we can construct a perturbation {φⁱ_n}_n of {ϕ_n} that is L^q−good for q ≥ p_i, and L^q−bad for 1≥ q < p_i. Consider a sequence of blocks {Bκ}κ∈N, where Bκ = {φ^κ_nκ−1+1, ..., φ^κ_nκ,} and {nκ} is a sequence of positive integers increasing to infinity. Let D_κ = {nκ−1 + 1, ..., n_κ}, and let {φ_n}_n = U_κB_κ. Now fix q > p. There exists n_o ∈N so that for all n > n_o we havepn< q,

∞

X

κ=no

X

n∈Dκ

(1−α^κ_n)^q ≤

∞

X

κ=no

X

n∈Dκ

1 (nlog²n)^q/pno

≤ X

n

( 1

nlog²n)^q/pno <+∞.

Using Proposition 1.4(c) we get φ_n ∗ f → f for f ∈ L^q(R), q > p, or ˆ ˆ

φ_n(ξ) ˆf(ξ)→f(x) forq > p.

Now consider a sequence C_i^N → ∞ as i → ∞. Since {φⁱ_n}_n is L^q−bad for allq < p_i, it is also L^p−bad. These existsf_i ∈L^p([0,1)) andλ^N_i >0 such that

|{ sup

n>ni−1

φⁱ_n∗f_i(x)}| >

Z

−Jκ

|φⁱ_n(x)f_i(y)|^pdy

> C^N kfi(x−λ^N_i )k^p_p

= 2C_i^N,[kf_i(x−λ^N_i )k_p= 2¹⁻ⁱ, C^N = 2^(i−1)p+1C_i^N].

It follows that there exists n_i > ni−1, so that

|{ sup

ni−1<n≤ni

(φⁱ_n∗f_i)}|> C_i^N. Set

f˜=X

i

fi, then kf˜kp≤X

i

kfi kp≤2.

Suppose that{φn}satisfies a weak (p, p) inequality inL^p([0,1)). We know that ifµbe a finite positive Borel measure, then these exists a sequence µ_n of atomic measure that converges to µweakly or if f has compact support then

Z

R

dµ_nf(x)→ Z

R

f(x)dµ or

µ_n→µweakly .

Iff ∈L¹(R), dµ=|f(x)|dx is a finite Borel measure, so we can find

µ_n =

N

X

ı=1

C_i^Nδ_λ^N

i →µ weakly.

Consider

|{sup

n

(φⁱ_n∗f)}| = Z

−J_κ

|φⁱ_n(y)f(x−y)|^pdy

≤ Z

−J_κ

|φⁱ_n(y)d_µn(x−y)|^pdy

≤ k

N

X

i=1

f(x−λ^N_i )C_i^N k^p_p

≤

N

X

i=1

C_i^N kf(x−λ^N_i )k^p_p

≤ C_o^N kf k^p_p

= 2^pC_o^N. (1)

On the other hand,

|{sup

n

(φ_n∗f)}| ≤ |{ sup

ni−1<n≤n_i

(φⁱ_n∗f(i))}|> C_i^N (2) Combining Equations (2.1) and (2.2) we get

C_o^N > C_i^N

But C_i^N → ∞ as i → +∞. Hence φ_n∗f 9 f in L^p([0,1)). Since the spaces L^q([0,1)) are nested, {φ_n} is L^q([0,1))-bad for all 1 ≤q ≤p. Therefore, such a choice of {n_κ} makes {φ_n}L^q(R)−bad for all 1 ≤ q ≤ p. This implies that

ˆ ˆ

φ_n(ξ) ˆf(ξ)9f(x) for 1≤q≤p.

(iii) Let {ϕ_n}n∈N be a good approximate identity, and let {ζ_n}n∈N be any approximate identity. Let {p_n} be a sequence of real numbers satisfying

1≤p₁ < p₂ < ... < p_n % ∞

Consider the blocks{B_κ}, where each blockB_κ is related top_κ.fori∈D_κ, let φ_i =α^κ_iϕ^κ_i + (1−α^κ_i)σ_i^κ.

Choose n_κ such that α^κ_i →1. Then since {ϕ_n} is L^∞ good, ϕ_n∗f →f a.e. for all f ∈L^∞(R), and

α^κ_iϕ^κ_i ∗f →f a.e. for all f ∈L^∞(R).

Since

σ_i^κ∗f(x)≤kf k∞ .

(1−α^κ_i)σ_i^κ∗f →0 a.e. for all f ∈L^∞(R).

It follows thatφ_n∗f →fa.e, for allf ∈L^∞(R).This implies that ˆ

( ˆφ_n(ξ) ˆf(ξ))→ f(x) forq =∞.

The approximate identity {φ^κ_n}_n is L^pm−bad for every m ∈ {1, ..., κ}, since it is L^q−bad for every 1 ≤ q ≤ p_κ. There exists f_m^κ ∈ L^pm([0,1)) with kf_m^κ(x−λ^κ(Nm ⁾)k= 2^−κ, λ^κ(N)m >0 and n^κ_m > mκ−1 so that

|{ sup

nκ−1<n≤n^κ_m

(φ^κ_n∗f_m^κ)}| > C^N kf_m^κ(x−λ^κ(N_m ⁾)k^p_p^m_m

= C_κ^N 2^κpm Let ˜f =P

κ≥κof_κ^κ_o, then kf˜k_pκo<2.

So

|{sup

n

(φ_n∗f˜)}| ≤ C₀^./kf˜k^p_p^κo_κo

≤ 2^pκoC_o^N. (3) Hence

|{sup

n

(φ_n∗f˜)}| ≥ |{ sup

nκ−1<n≤nκ

(φ^κ_n∗f_κ^κ_o)}|

> C_κ^N

2^κpκo (4)

using (2.3) and (2.4) we get

C_o^N > C_κ^N

2^κpκo(κ+1) →+∞

.Thus we conclude that ˆ ˆ

φ_n(ξ) ˆf(ξ)9f(x) For 1≤q <∞.

Hence the proof is completed.

References

[1] A. Bellow, Perturbation of a sequence, Advances in Mathematics, 78(1989), 131-139.

[2] K. Reinhold-Larsson, Discrepancy of behaviour of perturbed sequences inL^p spaces, Proc. Amer. Math. Soc., 120 (1994), 865-874.

[3] S. Sawyer, Maximal inequalities of weak type, Ann. of Math., 84 (2)(1966), 157-174.